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Hilbert s Mistake Edward Nelson Department of Mathematics Princeton University 1 Abstract Hilbert was at heart a Platonist No one shall expel us from the paradise that Cantor has created for us His formalism was primarily a tactic in his battle against Brouwer s intuitionism His mistake was to pose the problem of showing that mathematics beginning with Peano Arithmetic is consistent rather than to ask whether it is consistent In this talk I give reasons for taking seriously the possibility that contemporary mathematics including Peano Arithmetic may indeed be inconsistent 2 Potential vs Completed Infinity Let us distinguish between the genetic in the dictionary sense of pertaining to origins and the formal Numerals terms containing only the unary function symbol S and the constant 0 are genetic they are formed by human activity All of mathematical activity is genetic though the subject matter is formal Numerals constitute a potential infinity Given any numeral we can construct a new numeral by prefixing it with S Now imagine this potential infinity to be completed Imagine the inexhaustible process of constructing numerals somehow to have been finished and call the result the set of all numbers denoted by N Thus N is thought to be an actual infinity or a completed infinity This is curious terminology since the etymology of infinite is not finished 3 We were warned Aristotle Infinity is always potential never actual Gauss I protest against the use of infinite magnitude as something completed which is never permissible in mathematics We ignored the warnings With the work of Dedekind Peano and Cantor above all completed infinity was accepted into mainstream mathematics Mathematics became a faith based initiative Try to imagine N as if it were real A friend of mine came across the following on the Web 4 www completedinfinity com Buy a copy of N Contains zero contains the successor of everything it contains contains only these Just 100 Do the math What is the price per number Satisfaction guaranteed Use our secure form to enter your credit card number and its security number zip code social security number bank s routing number checking account number date of birth and mother s maiden name The product will be shipped to you within two business days in a plain wrapper 5 My friend answered this ad and proudly showed his copy of N to me I noticed that zero was green and that the successor of every green number was green but that his model contained a red number I told my friend that he had been cheated and had bought a nonstandard model but he is color blind and could not see my point I bought a model from another dealer and am quite pleased with it My friend maintains that it contains an ineffable number although zero is effable and the successor of every effable number is effable but I don t know what he is talking about I think he is just jealous The point of this conceit is that it is impossible to characterize N unambiguously as we shall argue in detail 6 As a genetic concept the notion of numeral is clear The attempt to formalize the concept usually proceeds as follows i zero is a number ii the successor of a number is a number iii zero is not the successor of any number iv different numbers have different successors v something is a number only if it is so by virtue of i and ii We shall refer to this as the usual definition Sometimes iii and iv are not stated explicitly but it is the extremal clause v that is unclear What is the meaning of by virtue of It is obviously circular to define a number as something constructible by applying i and ii any number of times We cannot characterize numbers from below so we attempt to characterize them from above 7 The study of the foundations of arithmetic began in earnest with the work of Dedekind and Peano Both of these authors gave what today would be called set theoretic foundations for arithmetic In ZFC and extensions by definitions thereof let us write 0 for the empty set and define the successor by Sx x x We define x is inductive Sy x 0 x y y x Then the axiom of infinity of ZFC is x x is inductive and one easily proves in ZFC that there exists a unique smallest inductive set i e x x is inductive y y is inductive x y 8 We define the constant N to be this smallest inductive set N x inductive x y x is inductive y y is and we define x is a number x N Then the following are theorems 1 0 is a number 2 x is a number Sx is a number 3 x is a number Sx 6 0 4 x is a number y is a number x 6 y Sx 6 Sy 9 These theorems are a direct expression of i iv of the usual definition But can we express the extremal clause v The induction theorem x is a number y is inductive x y merely asserts that for any property that can be expressed in ZFC if 0 has the property and if the successor of every element that has the property also has the property then every number has the property We cannot say For all numbers x there exists a numeral d such that x d since this is a category mistake conflating the formal with the genetic 10 Using all the power of modern mathematics let us try to formalize the concept of number Let T be any theory whose language contains the constant 0 the unary function symbol S and the unary predicate symbol is a number such that 1 4 are theorems of T For example T could be the extension by definitions of ZFC described above or it could be Peano Arithmetic P with the definition x is a number x x Have we captured the intended meaning of the extremal clause v To study this question construct T by adjoining a new unary predicate symbol and the axioms 5 0 6 x Sx 11 Notice that is an undefined symbol If T is ZFC we cannot form the set x N x because the subset axioms of ZFC refer only to formulas of ZFC and x is not such a formula Sets are not genetic objects and to ask whether a set with a certain property exists is to ask whether a certain formula beginning with can be proved in the theory Similarly if T is P we cannot apply induction to x since this is not a formula of P Induction is not a truth it is an axiom scheme of a formal theory If T is consistent then so is T because we can interpret x by x x And conversely of course if T is inconsistent then so is T For any numeral S S0 we can prove S S0 in S S0 steps using these two axioms and detachment modus ponens 12 Let d be a variable …


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